4x^2-15x+11=3x+6

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Solution for 4x^2-15x+11=3x+6 equation:


Simplifying
4x2 + -15x + 11 = 3x + 6

Reorder the terms:
11 + -15x + 4x2 = 3x + 6

Reorder the terms:
11 + -15x + 4x2 = 6 + 3x

Solving
11 + -15x + 4x2 = 6 + 3x

Solving for variable 'x'.

Reorder the terms:
11 + -6 + -15x + -3x + 4x2 = 6 + 3x + -6 + -3x

Combine like terms: 11 + -6 = 5
5 + -15x + -3x + 4x2 = 6 + 3x + -6 + -3x

Combine like terms: -15x + -3x = -18x
5 + -18x + 4x2 = 6 + 3x + -6 + -3x

Reorder the terms:
5 + -18x + 4x2 = 6 + -6 + 3x + -3x

Combine like terms: 6 + -6 = 0
5 + -18x + 4x2 = 0 + 3x + -3x
5 + -18x + 4x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
5 + -18x + 4x2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
1.25 + -4.5x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + -4.5x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + -4.5x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + -4.5x + x2 = 0 + -1.25
-4.5x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
-4.5x + x2 = -1.25

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = -1.25 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = -1.25 + 5.0625

Combine like terms: -1.25 + 5.0625 = 3.8125
5.0625 + -4.5x + x2 = 3.8125

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 3.8125

Calculate the square root of the right side: 1.952562419

Break this problem into two subproblems by setting 
(x + -2.25) equal to 1.952562419 and -1.952562419.

Subproblem 1

x + -2.25 = 1.952562419 Simplifying x + -2.25 = 1.952562419 Reorder the terms: -2.25 + x = 1.952562419 Solving -2.25 + x = 1.952562419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 1.952562419 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 1.952562419 + 2.25 x = 1.952562419 + 2.25 Combine like terms: 1.952562419 + 2.25 = 4.202562419 x = 4.202562419 Simplifying x = 4.202562419

Subproblem 2

x + -2.25 = -1.952562419 Simplifying x + -2.25 = -1.952562419 Reorder the terms: -2.25 + x = -1.952562419 Solving -2.25 + x = -1.952562419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -1.952562419 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -1.952562419 + 2.25 x = -1.952562419 + 2.25 Combine like terms: -1.952562419 + 2.25 = 0.297437581 x = 0.297437581 Simplifying x = 0.297437581

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.202562419, 0.297437581}

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